Homework 4 Solutions
Josh Hernandez
October 27, 2009
2.2  The Matrix Representation of a Linear Transformation
2.
Let
β
and
γ
be the standard ordered bases for
R
n
and
R
m
, respectively. For each linear transformation
T
:
R
n
→
R
m
, compute [
T
]
γ
β
b.
T
:
R
2
→
R
3
defined by
T
(
a
1
, a
2
) = (2
a
1

a
2
,
3
a
1
+ 4
a
2
, a
1
).
Solution:
A transformation from a 2 to a 3 dimensional space has a 3
×
2 matrix:
[
T
]
γ
β
=
2
1
3
4
1
0
.
c.
T
:
R
3
→
R
defined by
T
(
a
1
, a
2
, a
3
) = 2
a
1
+
a
2

3
a
3
.
Solution:
[
T
]
γ
β
=
(
2
1
3
)
.
4.
Define
T
:
M
×
2
×
2
(
R
)
→
P
2
(
R
)
by
T
(
a b
c d
)
= (
a
+
b
) + (2
d
)
x
+
bx
2
.
Let
β
=
v
1
=
1
0
0
0
, v
2
=
0
1
0
0
, v
3
=
0
0
1
0
, v
4
=
0
0
0
1
,
and
γ
=
{
1
, x, x
2
}
.
Solution:
T
(
v
1
) = 1,
T
(
v
2
) = 1 +
x
2
,
T
(
v
3
) = 0,
T
(
v
4
) = 2
x
. Thus
[
T
]
γ
β
=
1
1
0
0
0
0
0
2
0
1
0
0
.
5.
Let
α
v
1
=
1
0
0
0
, v
2
=
0
1
0
0
, v
3
=
0
0
1
0
, v
4
=
0
0
0
1
,
and
β
=
{
1
, x, x
2
}
,
and
γ
=
{
1
}
.
c.
Define
T
:
M
×
2
×
2
(
F
)
→
F
by
T
(
A
) = trace(
A
). Compute [
T
]
γ
α
Solution:
T
(
v
1
) = 1,
T
(
v
2
) = 0,
T
(
v
3
) = 0,
T
(
v
4
) = 1. Thus
[
T
]
γ
α
=
(
1
0
0
1
)
.
1
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d.
Define
T
:
P
2
(
R
)
→
R
by
T
(
f
(
x
)) =
f
(2). Compute [
T
]
γ
β
.
Solution:
T
(1) = 1

x
=2
= 1,
T
(
x
) =
x

x
=2
= 2,
T
(
x
2
) =
x
2

x
=2
= 4. Thus
[
T
]
γ
β
=
(
1
2
4
)
.
8.
Let
V
be an
n
dimensional vector space with an ordered basis
β
. Define
T
:
V
→
F
n
by the mapping
T
(
x
) = [
x
]
β
. Prove that
T
is linear.
Solution:
Denote
β
=
{
v
1
, . . . , v
n
}
. Given
v
=
c
1
v
1
+
· · ·
+
c
n
v
n
, and
w
=
d
1
v
1
+
· · ·
+
d
n
v
n
in
V
, and some scalar
a
,
T
(
v
+
aw
) =
T
(
(
c
1
+
ad
1
)
v
1
+
· · ·
+ (
c
n
+
ad
n
)
v
n
)
= (
c
1
+
ad
1
, . . . , c
n
+
ad
n
)
= (
c
1
, . . . , c
n
) +
a
(
d
1
, . . . , d
n
) =
T
(
v
) +
a
T
(
w
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 Spring '10
 SHALOM
 Linear Algebra, Algebra, Vector Space, cn vn, ndimensional vector space, aAik Bkj

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